Split-quaternion
In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain zero divisors, nilpotent elements, and nontrivial idempotents. As a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. The coquaternions came to be called split-quaternions due to the division into positive and negative terms in the modulus function. For other names for split-quaternions see the Synonyms section below.
The set forms a basis. The products of these elements are
and hence ijk = 1. It follows from the defining relations that the set is a group under coquaternion multiplication; it is isomorphic to the dihedral group of a square.
A coquaternion
has a conjugate
and multiplicative modulus
- .
This quadratic form is split into positive and negative parts, in contrast to the positive definite form on the algebra of quaternions.
When the modulus is non-zero, then q has a multiplicative inverse, namely q*/qq*.
is the set of units. The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •). The coquaternions with modulus qq* = 1 form a non-compact topological group SU(1,1), shown below to be isomorphic to SL(2,R).
The split-quaternion basis can be identified as the basis elements of either the Clifford algebra Cℓ1,1(R), with {1, e1=i, e2=j, e1e2=k}; or the algebra Cℓ2,0(R), with {1, e1=j, e2=k, e1e2=i}.
Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra.
Read more about Split-quaternion: Matrix Representations, Profile, Pan-orthogonality, Counter-sphere Geometry, Application To Kinematics, Historical Notes, Synonyms